Here's
one definition of old school
“The more of the following a campaign has, the more old school it is: high lethality, an open world, a lack of pre-written plot, an emphasis on creative problem solving, an exploration-centered reward system (usually XP for treasure), a disregard for "encounter balance", and the use of random tables to generate world elements that surprise both players and referees. Also, a strong do-it-yourself attitude and a willingness to share your work and use the creativity of others in your game.”
Here's
another:
Rulings, not Rules
Player Skill, not Character Abilities
Heroic, not Superhero
Forget “Game Balance.”
5e:
High lethality, forget game balance: I don't think 5e is set up very well for these concerns. Death saves and abundant healing (via spells and rests) make it more difficult for characters to die without a bunch of optional rules. Even then PCs will have more ways to mitigate lethal situations. You
can forget game balance to make things more dangerous, so I guess that's down to dm style, but it feels against the spirit of the game and the way it's typically played.
Player skill, not character abilities, emphasis on creative problem solving: this is hard because often times the best solution to a problem
is on your character sheet, or at the very least involves leveraging existing rules. Cantrips, first level spells, and racial abilities take care of most of your food, water, and light needs; keeping track of those is a hallmark of classic play. Encumbrance limits are extremely generous. An osr game like knave emphasizes randomly selected mundane equipment in order to spur creative use of items; this is less important in 5e, especially past level 3 or so.
Further, 5e is very skill-check heavy. Rereading the dmg yesterday, I noticed how many optional rules or procedures boil down to making 1 or 2 ability checks. And there are ways to get advantage and bonuses to those skill checks. In a large enough party, you can secure enough die rolls to succeed at most things.
Rulings not rules: 5e was obviously influenced by Finch's primer here. There are still quite a lot of rules in 5e though? But it is easy enough to handwave most things for a more fluid game. In 5e games turning to the rules usually happens when looking up spell descriptions. But what 5e lacks that earlier editions have are procedures, for example for
dungeon crawling.
An open world, exploration based xp: this is up to dm style, but seems less popular than playing structured adventures. In terms of xp, the main systems are either xp for combat (leading to combat as sport challenges) or milestone (for structured adventures), so not so much.
DIY attitude: honestly, alive and well! Yes, lots of people play wotc adventure paths, but they also produce loads of DM's guild content to
fix...erm, expand those adventures.
You can use a bunch of optional and house rules to mitigate some of the above, but I'm not sure it's worth it. That being said, my house rules to make 5e to be old school would involve: unconscious at 0 hp, death at -10; finite cantrip usage; spend HD to recover hp on long rest (or other slow healing rule); no or less darkvision; slot based encumbrance.