Approach Controller reported airspace incursion with adjacent Approach Control when automated handoff failed; citing old equipment and staffing as contributory.

Narrative

I was working the radar East sector and radar South sector combined. We had been holding aircraft for several hours because of traffic out to the east. There was turbulence; sometimes severe; in the area. Local started releasing aircraft and extra spacing was needed eastbound. Aircraft X was inbound from the east. I cleared the aircraft direct to the fix on the ZZZ arrival. I pointed them out to Radar West and attempted to flash the handoff to ZZZ1; the handoff failed. I was distracted by numerous other aircraft and answered them and took care of other separation. I then went back to Aircraft X and attempted another flash to ZZZ1; again it failed. Aircraft X was still approximately 15 miles west of the boundary and I intended to go back and manually hand the aircraft off but got very busy with other aircraft on frequency and did not get back to Aircraft X in time. I then accomplished a manual handoff to ZZZ1 Approach.At ZZZ Tower we have very old and antiquated equipment. We have been asking for STARS or anything better for over a decade. I have had several instances in the past where the ARTS system just seemed as if it could not keep up; i.e. dropped tags; any auto handoffs; spotty radar coverage; etc. I feel better equipment would greatly facilitate a smoother operation and allow more time to handle more traffic and unusual traffic situations. Also our staffing is running at bare minimums and most times there are no bodies to help out during busy times. I feel an extra set of eyes during a very busy period would have prevented this occurrence.