Are there any latching transistors without relays?
What you're looking for is called an SR latch. It's not a transistor - though it can be constructed from discrete transistors - and it's available in IC form, for example the CD4043/CD4044.

The latch has two input pins, (S)et and (R)eset. A high level on S turns the latch on, and a high level on R turns the latch off again.

More generally, if you're working on building your own processor, it's worth spending some more time reading about different types of latch and how they behave. D latches and registers in particular are likely to prove incredibly useful to you.

Latches are also frequently called Flipflops.