How accessible this actually is
Multiple ski resorts operate within roughly one to two hours of Seoul, several running dedicated shuttle buses directly from central departure points in the city — no car required, which makes this considerably more accessible to visitors than ski trips typically are in most countries. Resorts generally offer full rental packages (skis, boots, and outerwear), so arriving with no gear of your own is completely normal.
What to expect from the terrain
Korean ski resorts near Seoul are considerably smaller and gentler than major ski destinations in the Alps or Rockies — good for beginners and intermediate skiers, and for a fun day trip, but not the goal if serious advanced terrain is what you're after. Most operate a mix of day and night skiing sessions, with night skiing under floodlights being a distinctly popular and normal option here.
Should you build a trip around it
Worth it as a single day-trip add-on to a winter Seoul visit rather than the centerpiece of your trip — it's a genuinely fun, easy-to-arrange activity that most visitors don't think to consider, precisely because "skiing near a major East Asian capital" isn't the first association most people have with Seoul.