What is the volume of spacetime we can survey? How large is the fraction of spacetime volume we can observe within a sphere with the radius of the present particle horizon distance?
 A: According to Wikipedia, 
https://en.wikipedia.org/wiki/Observable_universe
even though the Universe is only about 14 billion years old, the present distance of the farthest objects whose light reaches us now is as far as 46.5 billion light years or $4.4\  10^{26}$ meters. This distance is just what is meant by the "particle horizon". The volume of the sphere with this radius is $4 \ 10^{80}$ cubic meters.
(Note that the number experessed in meters in Wikipedia is for the diameter, twice the radius, but the size in light years is for the radius. Not very logical !).
It is curious that the horizon should be more than 14 billion years away, but this is because the Universe is now in near exponential expansion, thanks of the dark energy that now dominates the content of the universe, and has been so for some time, but not for most of the time since the Big Bang (if the expansion had been exponential for much longer, the horizon could have been very much farther away than the age of the Universe times the speed of light). 
Anders Sandberg tells me your question is not about the size volume up to the horizon but what fraction  of this volume is observable, i.e. up to the last collision of photons from the cosmic microwave background. This is the fraction observable with photons, but if we develop neutrino based observations, or gravitational wave based observations, we could get much closer to the horizon.
Anyway, the same Wikipedia page gives 46.5 billion light years for the horizon, and 45.7 billion light years for the distance for the radius of the (photon based observations) observable universe. The cube of the ratio of the latter to the former is about 94%.
Is this indeed what you are asking ?
OK, so let us try to attack the problem you described in your comment below :
Well, Alfred's comment doesn't answer my question. To make things more clear, we know that at the present time our particle horizon consists of a spatial volume with a radius of about 46.5 Gly. Now, within this volume our past light cone is embedded. It is a surface that is rotationally symmetric about our world line. This surface is tearshaped (or tent-like shaped in comoving coordinates). It reaches towards the Big Bang. On that surface are located all light emitting sources we can observe today. The question can then be re-formulated: Does this surface have any (small) 3D thickness?
What you are considering is not a volume in 3D space, but a "cone" in 4D space. This is not the 4D "volume" expressed in cubic meters times second of Chen's suggestion in a comment to your question (not in a comment to this answer), but its "skin", a 3D object in two space dimension multiplied by length of time.
The 2D dimension, the area now is $4\pi R^2$ where $R$ is 46.5 billion light years or 4.4 10^26 meters. 
But this was smaller in the past. Though long ago, for a short time after the Big Bang, it was radiation dominated, the Univers spent most of its lifetime very close "dust flat universe", at least if the usually admitted "cold dark matter" is indeed correct, because the contribution of dark energy is constant while that of dark matter was considerably larger in the past. As Chen has pointed out, in that model the radius of the particle horizon is $3cT$ when the age of the Universe is $T$. According to Wikipedia the age of the Universe is 13.8 billion years. In the "dust flat universe" the radius should be 41.4 billion light years.
It is now somewhat (but not much) larger because the influence of "dark energy", which works like Einsteins's "cosmological constat"
Indeed the "dark energy" has only started to dominate "recently", contributing only just above 10% increase (46.5 instead of 41.4). The growth is far from being exponential yet (thoug it will become, when the matter density will contribute less and less. So the "flat dust universe" value should be still rather close to the exact value, to within a few tens percents. I am speaking now of the (2 space plus time) 3D object. I did the calculations, the result is 
$12\pi/7 R^2 T$ where T is the age of the Universe
Now if you want to express this in 3D volume, it may be reasonable to multiply by the speed of light. But is is only a convenience. It is not a real physical 3D volume, just a way to think of a (two space dimensions plus time) object as if it were a 3D volume. Since $T=T/3c$ it comes to $4\pi/7 R^3$. 
But the ratio of this object to Chen's measure of the 4D interior, $\pi R^4/9$ is not a fraction, but the inverse of a length. 
A: After studying the many answers (related or not - but many thanks anyway) to my question, I can now summarize my understanding: As the 3D (= 2D surface + time) past lightcone shell has zero spatial depth, its fractional extent within a 4D (= 3D space + time) spacetime object like the particle horizon "volume" must be also zero or infinitly small. -
Cosmic surveys can only map light-emitting events that lie on our past lightcone.
