How can I protect equipment plugged into ungrounded outlets?
You should ABSOLUTELY install GFCI protection. This can be done at the electric box by replacing the normal circuit breaker with a GFCI breaker or at the outlet using a GFCI outlet. Whatever else you do is up to you, but this is the bare minimum for safety. Ungrounded outlets without GFCI are now considered unsafe and substandard.

If you are a renter, you can probably make your landlord pay for it.

The primary purpose of GFCI is to open the circuit in the event of a ground fault current. It is very sensitive, and a small ground fault current will cause it to trip very quickly. It can probably protect equipment in some scenarios, but more importantly it protects you from injury, and protects your landlord (if applicable) from getting sued in the event of an electrical fault causing injury.