I want to design a simple circuit where pushing the push to "make" switch toggles the state 1 or 0 so that it remembers its last state.Without using a logic or flipflop ic is it possible to construct a circuit that does this out of transistors and how does it compare with a ic in terms of handling contact bounce etc ?

 

How about a 555? I designed something similar a while back, it will eventually be in the AAC book.

Submission: 555 Bistable Multivibrator

You can also make a simple set/reset flip flop using transistors and use the same feedback mechanism.

 

About 48 years ago logic ICs were made with only resistors and transistors (RTL) and they made some flip-flop circuits. Make one if you must.