How radioactive is uranium? Look at this video: 
People face uranium directly. Does this mean the radioactivity of uranium is very weak? Because its half-life is very long? Personally, I would never dare to touch any radioactive element. 
I also remember seeing people holding a big chunk of uranium in hand. See here
 A: There are two sides to this question.
Naively, the answer would be "bah, not much" because it is not terribly active and neither alpha, nor beta radiation is really dangerous. The former (which occurs early in the decay chain) is absorbed even by a few centimeters of air, and the latter (which appears later in the decay chain) is unable to penetrate the callus layer of your skin. The callus is dead tissue either way, so radiation doesn't really do anything to it.
However, uranium is directly toxic (nephro- and hepatotoxic, and causing neurological effects) and finally decays to an accumulating neurotoxic element (lead). The toxity is generally much more severe than the radioactivity. Uranium dust can very well be inhaled if no precautions are taken (not uncommon in fertilizer production).
But what's worst, your body happily absorbs uranium as "calcium" and puts it in your bone matrix.
Now, you will remember I just said alpha and beta radiators are pretty harmless. Alpha and beta radiators inside your body and especially near highly active tissue (such as certain organs, but also... bone marrow) are extremely harmful.
Further, if you look in the decay chain, you will notice quite a few elements appearing, some of which (radon) are gases which you can neither smell nor see but nevertheless inhale and absorb. Polonium... remember what substance it was the KGB used to murder Alexander Litvinenko?
Therefore, from a biological point of view, the answer must be: "very". You can certainly handle uranium safely with simple rubber gloves and behind a suction (or wearing a breath mask), but otherwise playing with it is not such a terribly good idea.
A: Natural uranium consists of $\approx 0.7\ \%$ $^{235}_{92}\mathrm U$, where the rest is $^{238}_{92}\mathrm U$.  Fresh reactor fuel consists of $3.5\ \% {-}4.5\ \%$ $^{235}_{92}\mathrm U$.  Both isotopes of uranium have very low specific activity and their radioactivity will by no means, under normal conditions, cause a higher dose than $20\ \mathrm{mSv}$, which is the annual limit dose for people working with radioactive materials (in the EU).  Uranium is, however, chemically toxic (as are all heavy metals). Therefore, it should not be consumed or handled with bare hands. The low specific activity $\mathrm{\frac{Bq}{g}}$ can be explained with the large half-life of the isotopes. This is best illustrated by the formula for calculating the specific activity
$$
A=\frac{N_\mathrm{A}\log(2)}{T_{\frac{1}{2}}m}.
$$ 
Therefore, large half-life $T_{\frac{1}{2}}$ results in very small activity $A$ per mass $m$.
It is a completely different question if the uranium has been irradiated. In this case, you would start building fission products and minor actinides, some of which are highly radioactive.  Handling them requires special equipment. As a rule of thumb,the larger the irradiation time (say in a reactor core) and the denser the neutron flux $\frac{n}{\mathrm{cm^2\ s}}$ the larger the radiotoxicity.
To summarize, fresh uranium fuel and natural uranium have very small specific activity. Anyway, I don't recommend playing with such materials because they are chemically toxic and you never know if the material has been irradiated. In radioactivity as well as in medicine it is all a question of dose.    
Remark: I got some questions about the equivalent dose form Uranium. Here is a simple (highly conservative) estimate. 
Suppose we had 1 kg of natural uranium. Natural uranium has specific activity of $\approx 2.6 \cdot 10^7\ \mathrm{\frac{Bq}{kg}}$. Here Bq means one decay per second and measures the activity of the source. Suppose further it emits ONLY gammas at $^{137}\mathrm{Cs}$ decay energy of $0.662\ \mathrm{MeV}$. Assume also that one somehow absorbed everything that is emitted by the uranium chunk. Plugging that into formulas gives
$$
1\ \mathrm{kg}\times 2.6 \cdot 10^7 \ \frac{\mathrm{Bq}}{\mathrm{kg}} \times 0.667\ \mathrm{MeV}\times 1.6\times10^{-13}\ \mathrm{\frac{J}{MeV}}\times 3600\ \mathrm{\frac{s}{h}}= 9.9\times10^{-3}\ \mathrm{\frac{Sv}{h}}
$$
This estimated dose rate of $9.9\times10^{-3}\ \mathrm{\frac{Sv}{h}}$ or $9.9 \space \mathrm{\frac{mSv}{h}}$ is higher than $0.4\ \mathrm{\frac{\mu Sv}{h}}$ by a factor of $1000$, which is the upper limit for the background radiation dose rate in Europe. In the US the annual limit amounts twice that value. So for one year one would would accumulate
$$
9.9\times10^{-3} \mathrm{\frac{Sv}{h}} \cdot 365 \mathrm{\space days} \cdot 24 \mathrm{\space hours} = 87 \mathrm{\space Sv}
$$
which is a lethal dose. 
Of course uranium does not emit only $\gamma$ radiation and you can't absorb all of it, unless you ate it, something I advised against. Moreover, you would spend only a limited amount of time near the material. Therefore, the dose you would get form 1 kg uranium will be much less than what I calculated. You can play with other energies, radiation types and exposure times. I chose $\gamma$ because it has the highest penetration depth and travels freely in air. Whereas, $\beta$ and $\alpha$ travel only short distances in air and are typically stopped by the skin or the clothes. Therefore, $\gamma$ is a quite conservative estimate. If the uranium emitted only $\alpha$ radiation and you absorbed it all the result will become $27$ times bigger. 
Another advantage is the high atomic number of Uranium, which makes it excellent gamma absorber. Therefore, significant percentage of the gamma rays will be absorbed by the source itself.
Moreover, as most radioactive heavy elements, the isotopes of urnaium would emit high energy alpha particles (energies about 5 MeV) and only low energy gammas. With the most energetic gamma line belonging to  $ $$^{235}_{92}\mathrm U$ having energy of $0.16 \mathrm{\space MeV}$. Low energy gammas are easy to absorb and have lower biological hazard factor. 
As with all alpha emitters, the most dangerous component is inhaling or ingesting the radioactive source. 
Since the source strength is determined by the specific activity, which has units of $\mathrm{\frac{Bq}{mass}}$, one can use the mass to scale to different amount of radioactive material. One gram under the above conditions would yield
$$
9.9\times10^{-3} \mathrm{\frac{Sv}{h}}\cdot 10^{-3}=9.9\times10^{-6} \mathrm{\frac{Sv}{h}}
$$
If one would assume a point source, the dose at distance $R$ can be found using the inverse square law. 
$$
\mathrm{Dose \space at \space (R=0) \space}\cdot \frac{1}{4\pi\mathrm{R}^2}
$$
Real values:
Point source of 1 gram of natural uranium at a distance of 1 meter yields 
$$
\mathrm{2 \cdot 10^{-12} \space\frac{Sv}{h}},
$$
which is much lower than the natural Background. 
 For this calculations the ICRP 72 conversion factors were used.  
Correction:
I used wrong initial conditions, I was talking about total activity, bit was taking only the gamma part of the source. Therefore, I have corrected the calculation.  
A: As with all safety hazards the answer is "It depends!". A chunk of natural uranium that hasn't been enriched or exposed to the inside of a reactor is not strongly radioactive and you can handle it with few precautions. You can hold it in your hand safely, but I would treat it the same way as I would treat all heavy metals that have a certain level of chemical toxicity. You do have to be concerned about toxicity if you are exposed through your lungs or to compounds: https://toxnet.nlm.nih.gov/cgi-bin/sis/search2/r?dbs+hsdb:@term+@na+@rel+uranium,+radioactive
If you are thinking about machining the metal or about chemical processing, I would suggest serious precautions and controls, as with any other substance that has even the slightest hazardous potential.  
The radiation is mostly low energy alpha and beta radiation which can't get through the skin to damage living cells... unless it's in the body, already, either through inhalation or by chemical absorption... same precautions as for the chemical poisoning problem. 
But here is the real problem: how do you know that what you are dealing with is a fresh piece of uranium that has just come out of the ground and that hasn't been exposed to neutrons? How do you know that it does not contain other radioactive contaminants which would have long decayed in a natural geological environment but which can be present in any amount if the material went through a processing facility that handles hot materials? Do you trust the friendly uranium dealer from around the corner who sold it to you? Really? What does he care about your health? I wouldn't trust that unless the material was properly tested by someone who can be trusted and I have an independent way of verifying that trust, i.e. at the very least I want to have a calibrated gamma/neutron monitor myself and a system to mark all materials that go through my possession in a reliable way. That's priceless. 
A: You wouldn't dare touching any radioactive element? So, you wouldn't eat, say, a banana? You are radioactive, as is pretty much everything you eat, and the ground where you live, and the air you breathe. Radioactivity is everywhere.
Most of the radioactivity in humans is from potassium-40, and a bit from radioactive carbon. Potassium-40 is more radioactive than U-238.
Of course, this is mostly a jab at your "I wouldn't touch anything radioactive". The popular understanding of radioactivity is dangerously bad, which is why people are afraid of nuclear fuel more than, say, the waste out of a coal power plant or of their own wood-burning furnace.
The main risks you have when handling something like a pellet of U-238 is:


*

*It can be ingested. Uranium is one of the more dangerous here, because it easily produces shavings that can move in the air and burn quite easily. U-238 decay mostly emits alpha radiation which is relatively harmless to humans, as long as you keep it outside. Needless to say, it becomes a lot more of a problem when it sticks to your lungs and gets into your blood (though that already poses extra problems due to it being a heavy metal - it's highly toxic regardless of its radioactivity).

*It's very concentrated - you're holding a big slab of radioactive material. The potassium in a banana is highly radioactive, but there's so little of it that it doesn't pose a real hazard.


As long as you keep your gloves on and isolate the air (as in the video), you'll be fine, especially if it's something you dug from the ground - danger from radioactive source is inversely proportional to lifetime of that source; uranium must necessarily have very little radioactivity, since it's existed as long as the Earth and there's still plenty to go around.
Don't mess around with those radiotherapy sources, though (warning: very much not pretty with a lot of "how could they be so stupid"). If you decide to read about that incident, note that even with the vastly more dangerous radioactivity source, the serious health issues (including amputation and death, sadly) were a result of a long exposure (many hours) and/or ingestion.
Needless to say, this shouldn't be taken as an advice to go ahead and play around with highly radioactive stuff. It is dangerous, just like, say, mercury is dangerous. It can kill you. All facilities dealing with highly radioactive matter have strict measures to prevent accident and measure exposure, and the gloves you see in the video aren't your typical household cleaning gloves. Different radioactive materials can have vastly different dangers, depending on their half-time and the emission characteristics.
