I think it's more common outside the United States. It's incredibly rare to have an employment contract here unless you're a high muckety muck.

Whether you realize it or not, you have one. Some companies might refer to it as your "Employment Agreement" rather than your "Employment Contract" - but it's still legally a contract even if they call it an 'agreement'. It is the sum total of everything that was negotiated and agreed upon when you accepted the position. Things like your starting salary, the amount of annual vacation you get, the sick pay/leave policy, agreements for annual bonuses or bonus modifiers, agreements for any stock grants or options rewards, stock option vesting policy and schedule (if applicable), whether your position is regionally bound to a specific region or location. In addition to all of the above, the state you reside in both when you accepted the position and where you live now (if it's different) impact your employment contract.

I also work for a very large company as a salaried employee and even though I started over 5 years ago, I can still download copies of all my original onboarding documents and forms, including my employment contract. My last company was just a small 35-person startup, but we had employment contracts there also. I still have my hardcopies of all that (including my required modifications).