In 5e's version of the flesh to stone spell, if the spell is maintained for the full minute duration, the effect (petrification) lasts until "removed".
How can this petrification be removed?
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Petrified is a condition, like sickened or poisoned. Many spells can remove harmful conditions — Greater Restoration lists petrification. It is currently the only one, short of Wish.
Wish can either emulate Greater Restoration directly (its "duplicate a spell" functionality), or as part of a greater effect healing all hit points and removing all conditions that Greater Restoration can, on up to twenty creatures all in one go (at some cost to the caster).
The 5e Monster Manual notes that Basilisk stomach acid is useful in potions of stone to flesh so… there's that. It's just a narrative hook at the moment, but perhaps the DMG will flesh that out further (or maybe they'll just leave that as an exercise for the DM, which would be fine since it's pretty easy to guess what that potion does).
Petrified is a condition, there are currently two spells that remove it. Greater Restoration as mentioned here, and Wish, the 9th level spell.
Usually there are also consumables (potions/salves/oils) that will remove petrification, however we have not seen any of those yet (we'll likely see them with the DMG though). You can also make it a plot element to find someone who can remove petrification.
Outside of the Restoration and Wish spells, and the acid from a Basilisk stomach, there is also the cockatrice feathers that can be transformed to a balm that you apply on a person who was petrified within the last 24h (MM p. 42).
The main problem here is... you need to find the cockatrice within those 24h because the balm has to be made from fresh feathers. So this is probably not an easy one to perform (especially if you have to carry the statue of your friend without breaking it...)
Maybe if you were to capture a live cockatrice and keep it around while you are adventuring.