Is any extra equipment necessary for strength training?
No.
lol short post today...
Necessary? No, but useful? Very.
The most efficient way to get stronger is to use incrementally loadable, ergonomically articulable, instruments like a barbell. If you're going to train with a barbell, there are tools that make that safer and easier.
In order of importance...
Squat shoes, with perfectly flat bottoms, Velcro straps across the top, and a slightly elevated heel will increase the quadriceps' capacity to help in the squat while also preventing force loss through squishy soles.
Lifting belts, with uniform widths and thicknesses, will generate hoop tension that will contribute to intra-abdominal pressure doing the movements, allowing you to better utilize your abdominal musculature and keep your back safe.
Wrist wraps will squeeze your hands when they flex or extend too much, giving you excellent intra-set feedback about how neutral your wrist angle is during the pressing movements.
There are a lot more options, but if these three are the only pieces of equipment you get, you can extract a lifetime's worth of training without missing a beat.