If you couldn't get your hands on a timepiece with a second hand, the utility of seconds would seem limited to the (non-coincidental) fact that they are about the same as a resting heartbeat period. Years and days might seem less arbitrary to us, but we would have trouble convincing our friends on Tau Ceti IV.3.3 Remember, our perspective in Physics is universal, and in that perspective all units are arbitrary.
We choose all our measurement conventions for convenience, often with monumental short-sightedness. The decimal number system is a typical example. At least when we realize this we can feel more forgiving of the clumsiness of many established systems of measurement. After all, a totally arbitrary decision is always wrong. (Or always right.)
Physicists are fond of devising "natural units" of measurement; but as always, what is considered "natural" depends upon what is being measured. Atomic physicists are understandably fond of the Angstrom (Å), which equals 10-10 m, which "just happens" to be roughly the diameter of a hydrogen atom. Astronomers measure distances in light years, the distance light travels in a year ( m), astronomical units (a.u.), which I think have something to do with the Earth's orbit about the sun, or parsecs, which I seem to recall are related to seconds of arc at some distance. [I am not biased or anything....]
Astrophysicists and particle physicists tend to use units in which the velocity of light (a fundamental constant) is dimensionless and has magnitude 1; then times and lengths are both measured in the same units. People who live near New York City have the same habit, oddly enough: if you ask them how far it is from Hartford to Boston, they will usually say, "Oh, about three hours." This is perfectly sensible insofar as the velocity of turnpike travel in New England is nearly a fundamental constant. In my own work at TRIUMF, I habitually measure distances in nanoseconds (billionths of seconds: 1 ns = 10-9 s), referring to the distance (29.9 cm) covered in that time by a particle moving at essentially the velocity of light.3.4
In general, physicists like to make all fundamental constants dimensionless; this is indeed economical, as it reduces the number of units one must use, but it results in some oddities from the practical point of view. A nuclear physicist is content to measure distances in inverse pion masses, but this is not apt to make a tailor very happy.