Another great way to take a set beyond muscle failure when training home alone is to do partial reps. Partial reps simply mean that you do the exercise over only part of the range of motion (ROM). Muscle failure is the inability to complete the rep with good form. That means your muscle fails to complete the rep over the full ROM on its own. It does not mean that the muscle has reached absolute failure and cannot perform any movement at all. This means that the muscle can move the weight over some of the ROM of the exercise.
For example, when doing dumbbell shoulder presses you reach failure when you are unable push the weight all the way over head. However, your delts can still pull press the weight up a little bit of the way. So, after reaching failure on presses, you continue doing reps through as much of the ROM as possible until even the slightest ROM becomes impossible. At that point you have taken the muscle to absolute failure.
Partials are great to use on bench presses (barbell and dumbbell) and flyes for chest, overhead presses and upright rows for delts, pull-ups, pulldowns and rows (barbell, dumbbell and cable) for back, curls (barbell and dumbbell) for biceps, pressdowns and overhead extensions for triceps and squats for legs.