I suppose this is a not-too-bright question but, in a nutshell: why are nuclear fuels radioactive?

With this I mean, which is the connection between being a fissile (or fertile, for that matter) material and being radioactive? Does the former imply the latter? Does the latter imply the former? Why?
Would it be thinkable for an isotope to be able to capture a neutron, become instable and split without it being radioactive in the first place?

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I think that is actually a good question - it's something that I've wondered about myself in the past, though I don't remember what I learned about the answer. –  David Z Apr 8 '11 at 20:02

Qualitatively, isotopes that carry a lot of unnecessary extra energy are unstable - and it makes them more unstable with respect to alpha or beta decay as well as fission (the latter is useful for nuclear fission energy).

However, the question why fissiles are radioactive (unstable) has a trivial answer. Only elements up to $Z=82$, $N=126$ i.e. $A=208$ may be stable nuclides, see

http://en.wikipedia.org/wiki/File:Table_isotopes_en.svg

On the other hand, fissiles are heavier - with $A=233,235$ (uranium), and $A=239,241$ (plutonium). Obviously, all those nuclides must be unstable - even though the half-life may be close to billions of years (very long).

In most cases, it's much more favorable for a nucleus to alpha-decay than to undergo fission because the alpha particle has a very high binding energy per nucleon, around 7 MeV, which is close to the maximum value around 8.8 MeV per nucleon achieved by nickel and iron:

In particular, a fission to fragments with $2<Z<30$ is almost impossible because those have a relatively low binding energy per nucleon. If one can energetically win by fusion, it's pretty much guarantee that one may also find an energetically favored alpha decay because the alpha particle has a pretty high binding energy per nucleon, allowing the other decay product to have a similar binding energy per nucleon if not lower, thus making the decay more likely.

The opposite implication obviously doesn't hold: most of the known radioactive isotopes are the light ones that can't split by fission (alpha decay is not counted even though it is a sort of "minifission"). In general, alpha and beta radioactivity is much more common among the known nuclides than fission.

It's important to realize that the link between fissiles and radioactivity is a bit indirect: the main isotopes that bring radioactivity to spinach etc. and that represent a public health concern are not the heavy nuclei themselves: it's the decay products such as the isotopes of iodine, cesium, strontium etc. Sometimes they're products of long chains of radioactive decay; the fission of the heavy nuclei is just the beginning of the chain, a tip of an iceberg.

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Thanks, things are clearer now! –  NuclearIgnoramus Apr 9 '11 at 8:10