I love sandboxes even though I have only a limited experience running them. I am not even sure if what I call 'sandbox' is the same thing everyone has in mind... for me a sandbox campaign is characterized by the following elements:
1- A regional map: large enough so that the players have the feeling they can seek adventures for a fairly long time, but definitely not a whole continent which may feel too dispersive. A good reference could be one single region in the Forgotten Realms, as in past editions books (e.g. "The Shining South" or "The Silver Marches"). Unlike story-driven campaign, I think it's important to show the region's map to the players at the beginning, maybe even get a poster and make it visible all the time when playing. A great map has a variety of "points of interest" which corresponds to possible adventures: anything like a haunted forest, an abandoned castle, a rumoured tomb that are the topics of local folklore, and therefore PCs from the region already know about and can visit.
2- Level range: the starting region must feature adventures and encounters of a variety of levels in order to deliver the feeling that the world is not designed for the PCs. However it doesn't have to be without a range, the DM can decide for example that everything within the starting region is in fact 1st-5th level. Yes, there can be also the lost tomb of the 20th-level lich somewhere if you want, the general idea is not to blow all your weapons in the starting region, so that the players get the feeling that other regions farther away will feature even more dangerous adventures. It is a good idea to highlight "level zones" within your sandbox e.g. the haunted forest might be as a whole notoriously more dangerous than everywhere else, but also whole regions beyond the starting one (think World of Warcraft); if you don't mind a bit of metagaming, you can make the levels known to the players to make sure they know where they're going, and assume the characters are capable of knowing the general degree of danger even if they have no concept of 'level'.
3- Impossible encounters: once again, for me it is quite essential for a 'sandbox' to carry the feeling that the PCs aren't special (yet), and impossible encounters are possible (pun intended). How should the DM handle them? First of all, remind the players that impossible doesn't mean unavoidable. Second, that killing a monster doesn't always have to be an encounter's goal, things such as stealing an item or a prisoner from it, or even simply fleeing from it are perfectly viable goals; just don't put the PCs into a "Kobayashi Maru". But consider also being open about an encounter's difficulty: you can keep it within the narrative by telling the players what their characters know about monsters or with good description of what happens during combat, or you can flat-out tell the players if they still don't get it (don't be afraid to break OoC, I'd rather metagame with them a bit than cause a TPK and tell them "you should have figured it out").
4- The network of quests: the premise of a sandbox is that the PCs are actively searching for adventures, but blank freedom can lead to paralysis. Having lots of quests interconnected by clues, hooks, NPC's requests, found maps etc. keeps the party moving. Ideally, I wish to have at least always 3 leads available, so that they never have to wander blindly but they also always have some choice on what to follow up next.
5- Slow large-scale events: things such as a war with a neighboring kingdom, a villain's plot to take over the world, shifts in deities' affairs and general world-scale threats are better left in the background. Sudden changes and threats immediately remove attention from local errands, and that's when the sandbox feel is compromised. Take advantage of your sandbox campaign to avoid for once the cliche of saving the world with a bunch of combats, and instead try to deliver a sense that dread things grow slowly in the dark. Maybe the orcs army never really invades, but it's always there on the other side of the mountain. And have the big villains escape often, and come back every time.
6- Slow level advancement: if not really slow, at least not too fast... the sandbox feel implies you don't get quickly so powerful that you have to move out of the sandbox itself to get some challenge.
Other things are typical but perhaps not strictly necessary. Random encounters are quite a staple in sandbox campaigns, but you can in fact always plan instead of rolling. I think randomness is a tool for the DM who doesn't want to always make decisions, but instead enjoys herself seeing the fantasy world 'come alive' on its own. Just make sure that the encounters make sense within the location, i.e. if you're in a jungle region then have jungle-appropriate encounters (unless you occasionally want to purposefully make the players jump: "what the heck is a yeti doing in a tropical jungle? someone must be behind this!").
Resource management (food, water, money...) and assets management (place of residence, family ties, social roles and obligations...) are great additions, but somehow I don't feel they have to be a must. You can fade them to the background if the players aren't interested, and still not lose the general sandbox feel.
The other thing I don't find necessary is the idea of a gridded (hex or square) map. There's a certain nostalgic charm in a hex regional map, with possibly a different surprise to discover under each hex, but IMHO it's very optional, and I feel more at ease with a gridless map.
