The reason jetpacks never really caught on is based on the
confluence of
technical issues. First of all, we live on the
Earth. That's a 1-Gee
gravitational pull and folks that's a lot of gravity. Almost all
aircraft fly using
aerodynamic lift. Jet packs fly using thrust. While
rockets capable of supporting a human being have existed since the days
Robert Goddard, they burn lots of fuel, more than can efficiently be carried by a human. It's not like they can shut the motor off and glide for a while. And if the motor has a failure, the thing crashes, probably killing the
pilot.
The second issue is steering and stability. NASA built one back in the fifties, and it worked. Barely.
The key issue is that most aircraft steer themselves and control their attitude by moving some aerodynamic surface, such as the synchronized use of rudders and aileron. The shape itself can offer stability. But a jet pack with enough control surfaces to fly that way is no longer a jetpack, it's an ultralight with the pilot as the landing gear. People got hurt. The stability issue was partly tamed with the use of gyroscopes, but they were almost impossible to fly. Taking off and landing again proved all experienced pilots could manage, nothing at all like Dave Stevens' character, The Rocketeer. The NASA program ended long before the modern computer era, so a fly-by-wire design should work better. However, it would still use the attitude jet control system, which means steering would consume even more fuel.
The third problem was bulk. Rocket exhausts tend to be hot, and it would not be good to char grill your intrepid young birdman. NASA was forced to move the flight jets away from the pilot. The attitude jets were also spread out for greater efficiency, so the old system was about six feet wide.
The combination of these factors makes jetpacks impractical for the foreseeable future. They were expensive, too large and heavy to carry around, and offer too little maneuverability and endurance to be worthwhile. Other systems work better. So we can build one, but the question is why?