Is there an issue keeping NFA items in an apartment and not locked in a safe?  I recently got a small place temporarily and would like to have my SBR with me, but can't move a safe here.

There are no federal storage requirements for NFA items.

There are small lockable safes, I had one and it was only like $299-399.00...  It would fit in a closet with room to spare.

I would stack several wooden crates of that Russian 7.62x39 in the bottom and a few Dumbbell plates on each side of those to give it enough weight so most thugs looking for a quick smash and grab hopefully wouldn't mess with it.  
It had room for 8-10 rifles no problem and a few pistols in the top of if.

When I moved and bought my current house, I emptied it and took it.  Now it just stores Ammo and a few of my dads Black Power Rifles.  
I have a much bigger safe for my guns, precious metals NFA paperwork and some files. I actually want a 3rd safe but my basement is fairly full crap and I have no good spot for it.

No problem, I've done it. Well I did have it locked up though. Small safe or case or something.

I have NFA items and I live in an apartment. My suggestion is keep them in a safe so your apartment management can't access.

+1

Apartment managers have access to the apartment.

Cable lock through mag and ejection port to something immovable ? Not a safe but better than nothing.

If you decide on something besides a safe, most hardware stores have some kind of tamper resistant screws. If you go with a cable lock, or wall mount, or something, get some large screws that can't be backed out easily and run them into studs.