How long should a half-wave dipole be?

Approximately L(m) = 142.5 / f(MHz) for thin wire (velocity factor ≈ 0.95). For example, at 14.1 MHz a dipole is about 10.1 m tip-to-tip, or ~5.05 m per leg.

What is the velocity factor?

Wire antennas are slightly shorter than the ideal electrical length because of the end effect and the wave travelling a touch slower along the wire. A factor of ~0.95 is typical for bare wire; insulated or thick conductors use a lower value.

Why is a quarter-wave monopole half the length of a dipole?

A monopole over a ground plane mirrors itself electrically, behaving like half of a dipole. So its physical length is λ/4 — half of the λ/2 dipole at the same frequency.

Are these lengths exact?

They are practical starting points. Wire diameter, insulation, height above ground and nearby objects all shift resonance, so cut a little long and trim while measuring SWR.