One mile is exactly 5,280 feet. The number looks arbitrary because it is: it is the residue of two incompatible measuring traditions being forced to agree by an act of the English parliament in 1593.
About the units
The mile descends from the Roman mille passus, a thousand double paces, which was about 4,860 modern feet. The furlong descends from something entirely different — the length of a furrow a team of oxen could plough before resting, standardised at 660 feet, or 40 rods. English land was measured in furlongs and rods, and English distance was measured in miles, and the two systems did not divide evenly into each other. The statute of 1593 resolved it by redefining the mile as exactly eight furlongs rather than adjusting the furlong, which lengthened the mile from its Roman value and produced 8 × 660 = 5,280 feet.
The exact factor
The factor is exactly 5,280 in one direction and 1/5,280 in the other. In metric terms a mile is exactly 1,609.344 m, a number that terminates because the foot is exactly 0.3048 m and 5,280 × 0.3048 works out cleanly. Related exact values worth having to hand: a furlong is 660 feet, a chain is 66 feet, a rod is 16.5 feet, and 80 chains make a mile. The chain in particular still appears in land records and in the layout of American cities, where blocks were often surveyed in chains.
Where you meet this conversion
Aviation is the biggest practical user of both units at once, since horizontal distance is in nautical or statute miles while altitude and runway dimensions are in feet — a runway is described as 10,000 feet long, which is 1.89 statute miles. Athletics uses the mile as a race distance with splits measured in metres. American road engineering specifies sight lines, gradients and lane widths in feet along routes measured in miles, and land surveying still carries the furlong and chain in older deeds.