hi lance this is an interesting and somewhat amusing question but agree its poorly formulated in its current form and Im rather surprised its at 26 votes currently.
but Ill take another stab/risk at an answer hoping that scoring will take the original difficulty and basic/inherent "soft" ambiguity of the question in mind and that based on current literature knowledge there are several possible avenues but arguably no "correct answer".
the main query seems to be "physics analogies in computer science" of which volume is one of them.
therefore its highly related to this other question
Physics results in TCS?
to answer this question Ill take a few different approaches which I think all have merit.
first, one approach sometimes used in physics and engineering fields is "dimensional analysis".
in this case strictly interpreted, volume is in the unit "space" or "length cubed". (although note in physics sometimes the term "space" is measured in either terms length or length cubed.)
therefore one could argue that in algorithmics the study of the space hierarchy is analogous to the study of "volume". a $O(n^3)$ space algorithm seems also to have the units "length cubed" where length is expressed in terms of input size.
the problem with this is that in algorithmics a $O(n^3)$ algorithm is merely a contrived limit, in a sense TCS has considered the "multidimensional space continuum" since its inception seen in the space hierarchy theorem.
carrying this over to physics, it might seem absurd to consider anything other than 3-d space, but of course advanced physics takes into consideration all kinds of higher dimensional space realms all the time, via c-dimensional hilbert [vector] spaces. so it would seem that in some sense an $O(n^c)$ space algorithm can be construed to operate in a c-dimensional hilbert space. seems like mulmuleys GCT program would be a premiere example of this.
another approach to a volume analogy (and other physics quantities) in TCS is as follows, as discussed in the other question. it is known that SAT has a transition point extremely analogous to the transition point in physics/thermodynamics, which happens eg with ideal gases under compression from one phase to another eg gas to liquid. this happens under a decrease in volume (say of the container of the gas). now in SAT with random inputs the main two parameters on input size are clauses and variables. (another parameter is number of variables in clauses, although that is often fixed at 3 for 3-SAT.)
adjusting either the clauses or variables while keeping the other fixed pushes the problem difficulty through the easy-hard-easy transition point. therefore it seems that these parameters are somehow analogous to Volume although I havent seen the specifics mapped out. digging into some of the deep papers on the statistical physics of SAT may turn up the analog of Volume. see  for a basic mapping of SAT onto statistical physics terminology.
 Analytic and Algorithmic
Solution of Random
Satisﬁability Problems by Mezard, Parisi, Zechina
the other approach might be related to the prior responses talking about high density information processing in physical space leading to black holes. it would use the plank limit which has units of time, space, mass etcetera. there is a plank Length unit $l_p$ defined on that wikipedia page. see units in this table
I dont know the exact physical interpretation of the plank length (its arguably an active area of physics research), but I believe it can be interpreted in some sense as "smallest possible length of space". in other words $l_p^3$ might be the size of a smallest possible physical bit. it is interesting and perhaps a discontinuity of the analogy that in physics a smallest bit apparently must actually have volume (ie units length cubed) in contrast to computer science in which bits seem to have a dimension of unit "length" only.