Abstract it out or approximate the simplest way you can
Around the tables I have played at, we dealt with it with heavy dose of approximation.
Most of the time, what the player really needs is "is the target too far for my spell/bow", the GM should use this to his advantage to adapt the distance. As a GM, part of the job is to to adjust the situations so that the mechanics work the way you want. Distance is just another variable.
In the case of feet to meter transition and precise distance, most games I've played used a grid approximation. When a better approximation was needed,we converted 5ft to 1m. Sometime adding a loose approximation of half the distance to get a better precision.
In combat with a grid, we would count squares for moves and range. So walking walking distance was measured as 6 squares instead of 30ft.
In combat without grid, we would count distance as multiple of a turn's walking distance. A target is either in melee (<10ft), in walking range (<30ft), charge range (<60ft), bow range (<160ft if memory serves) or not in combat (anything more than that)
For travel and distances between places in the world, the GM would express it in time at walking speed. So that a given point of interest would be at "3 days of walking" or "A day by boat". Inconsistencies between modes of transportation or different journeys were lampshaded respectively via different paths or changing conditions.
In roleplay scenes, distance ended up being abstracted.
Instead of "How far is the cook from me?", we would ask the GM :
Again, if we want a mental image of how far things were, divide by 5 and we have the In-universe distance.
As for why the imperial system is still in use, I would guess that they keep it because it's always been that way. But when the first editions came out, the US didn't use the metric system and the UK (see KorvinStarmast's comment below) had not yet widely accepted it.