Issue
I came across an HP Compaq Small Form Factor (SFF) Desktop PC which could not boot up and instead gave out 4 beeps, a pause, then 4 beeps again with a red blinking light at the power button. The beeping sequence was perpetual.
Solution
I opened the chassis and took a cleaning duster to blow out any dust build up I could find. A neat guide on how to do this properly can be seen here (I also wear an anti-static wrist strap as a precaution). While it seemed certain that it was an airflow problem (there was a lot of dust surrounding the CPU heatsink..) I rebooted but got the same results. So I took out all RAM sticks, used the cleaning duster to clear out any dust that may have gotten into the RAM slots after I took out the sticks and reseated them accordingly. Powered back on and it booted up just fine.
Conclusion
In short, it was the RAM sticks that simply needed to be taken out and reseated but still felt it was a good idea to clean out the dust build up that was getting to be pretty thick and could have potentially caused heating issues in the future.