If you're going to be there for awhile, I would get something as sturdy (read heavy) as you think your floor can handle and you can get up the stairs. The weight depends not only on size but the safe's construction (wall and door thickness). If you're living in a somewhat modern apartment and there is no significant termite or dry rot damage (you might not be able to tell though - about any damage), I would think the floor should be able to handle a three to four hundred fifty pound safe fairly easily (imagine you and a couple of friends standing in the same spot on the floor). Since we're not talking high end very heavy safes, Costco and similar stores often have decent deals on gun safes. I bought a Cannon 24 gun, 30 minute fire rating safe from Costco. They make a whole line of safes, from light weight to bomb proof. Some question Cannon's commitment to the customer but I'm happy so far. It weighs about 450 pounds and my son and I got it up one flight of stairs using an old rusty commercial dolly and a lot of elbow grease. A few friends would make the job pretty easy. It is lag screwed to the floor joists. It's fairly light for its size because the sheet metal used for its construction is probably 14, possibly 12 gauge. So a maniac with an axe (or a reciprocating saw could get through it with some time and noise. A nice higher end (and much heavier) safe would probably have 1/8" to 1/4" plate. Mine has a good lock and bolt mechanism and a sturdy door and internal hinges.
A good cabinet that is well secured to the floor and walls would be a lot better than nothing. You'd want a decent lock that can't be knocked off in 30 seconds with a hammer or cut off with bolt cutters. I'd go ahead and bolt it to the floor and possibly walls if you can find the joists and studs. Holes are easy to patch when you move and it will definitely afford you more peace of mind. An acquaintance of mine had two 750 pound safes (and a lot of valuable guns) carted right out of his house because they were not bolted down. It is more time consuming and a lot noisier to cut a safe loose from the floor and or walls. Depending on how much was going in the safe, maybe a smaller, heavier walled safe would work. But then again, I've always heard "buy bigger than you need now because you will fill it up". Sorry for all the verbiage and good luck.