Laser-plasma acceleration is an emerging technique for accelerating electrons to high energies over very short distances. The accelerated electron bunches have femtosecond duration, making them particularly relevant for applications such as ultrafast imaging or femtosecond X-ray generation. Current laser-plasma accelerators are typically driven by Joule-class laser systems that have two main drawbacks: their relatively large scale and their low repetition-rate, with a few shots per second at best. The accelerated electron beams have energies ranging from 100 MeV to multi-GeV, however a MeV electron source would be more suited to many societal and scientific applications. Here, we demonstrate a compact and reliable laser-plasma accelerator producing high-quality few-MeV electron beams at kilohertz repetition rate. This breakthrough was made possible by using near-single-cycle light pulses, which lowered the required laser energy for driving the accelerator by three orders of magnitude, thus enabling high repetition-rate operation and dramatic downsizing of the laser system. The measured electron bunches are collimated, with an energy distribution that peaks at 5 MeV and contains up to 1 pC of charge. Numerical simulations reproduce all experimental features and indicate that the electron bunches are only $\sim 1$ fs long. We anticipate that the advent of these kHz femtosecond relativistic electron sources will pave the way to wide-impact applications, such as ultrafast electron diffraction in materials with an unprecedented sub-10 fs resolution.