It is possible to go there, but your options are rather limited and expensive.
1: Get a job there! The meteorological station on Jan Mayen hires people (for 6 months at the time if I remember correctly), but you might need to be a Norwegian to apply (check out this website for details of when jobs are published, although none are available as of this writing (in Norwegian): http://jan.mayen.no/ledige-stillinger/)
2: Sail from Iceland: It is possible to sail from "nearby" areas, like Iceland. However, since Jan Mayen is a protected area you need a special permit to go there in motorised boats (and I assume this applies to sailboats with an engine as well). Check out this video for some inspiration from someone that did this in 2008: https://youtu.be/GsAsGTVy_8M
3: They have an airport! And that's pretty much the end of the good news. Only Norwegian military airplanes are allowed to land there (as well as emergency landings). And even if you managed to get permission to join them, they only fly every two months (one departed 4th of October and the next one is scheduled for 6th of December, fascinating how much information is publicly available online!).
4: Go on a cruise. This is by far the easiest option. You can get a trip for around 2000 euros, like this one: https://oceanwide-expeditions.com/the-arctic/cruises/otl02a-18-fair-isle-jan-mayen-spitsbergen-whale-safari, but they only depart a few times a year.
Last, I would say that going to Jan Mayen for the sake of having been to a very remote place is going to cost you a lot of effort and/or money. Norway offers some other very remote locations that are far easier to reach. I would recommend visiting Pyramiden, being the second northernmost settlement in the world and with a population that can be described with a single digit. You can fly from Oslo and Tromsø to Longyearbyen (which in itself is a fascinating and remote place) and take a boat in the summer or snowmobile in the winter to Pyramiden from here. It is an abandoned Russian mining town (now only occupied but a few people running a small hotel) and some polar bears.