I had $160 stolen from me after withdrawing it at an ATM with my PayPal card (which is linked to my Wells Fargo checking account). I was momentarily distracted and the funds were swiped from the tray in just a few seconds. Is it possible to get my funds replaced by either PayPal or my bank?
You should report the theft to the police. That is what it is. The police may also want to contact the ATM owner to get access to any security footage.
The ATM owner would only have responsibility if they provided substandard security. But based on all the locations I have seen ATMs I don't expect that there is any standard for security.
If as part of the incident the card was also stolen, then report that to the card issuer.
Depending on where you are it might be a loss you can deduct from your taxes, but the amount is probably too low. There is also the possibility of an insurance claim, but it would depend on the policy you have and any deductible that policy has. If you are going the tax or insurance route you should also get a copy of the police report for proof.