The observable universe has a size, the cosmic microwave background is what we 'see' at the 'edge' in terms of photons (~400k years after the big bang). We could see further if we could map out the gravitational wave or neutrino backgrounds (1 sec after the big bang).
But for now we can't really say if the universe in its entirety has a finite size.
For the gravitational wave background, maybe with LISA we might be able to get a glimpse, but the neutrino background seems like it'd take some truly unprecedented breakthroughs in our ability to detect neutrinos to have any chance of mapping it out.
Funny, in reading up on both, I had higher hopes for the gravitational waves.
It seems like GWB is a superposition of infinite overlapping waves that would be impossible to single out and "unwind" in order to form a map.
And big bang neutrinos are very weak, which makes them undetectable. My assumption was we'd need a breakthrough in measurement sensitivity but is there more to it?
But for now we can't really say if the universe in its entirety has a finite size.