If I pull on a metal bar with a small constant force forever, does it eventually break?

Worth adding the phrase in the first answer that is doing a lot of work: barring any other failure mechanism that occurs first.

Over the timescales where creep at low stress and low temperature matters, the bar will almost certainly fail by something else. Corrosion, fatigue from any vibration at all, damage, or the building around it being demolished.

That is not evasion. It is how engineering treats time-dependent failure generally: you calculate the design life and check that every mechanism has a comfortable margin over that life, rather than asking whether something lasts forever. Nothing does, and the useful question is which process wins.

Creep genuinely is the deciding mechanism in some designs — turbine blades, boiler tubes, anything hot and loaded continuously — and those components are designed to a creep life explicitly.

The crucial variable is temperature, expressed relative to the material's melting point rather than in absolute terms.

Creep becomes significant somewhere above roughly a third to a half of the melting temperature. For steel that is well above anything at room temperature, so a steel bar at ordinary conditions creeps immeasurably slowly. For lead, whose melting point is low, room temperature is already a substantial fraction of it — which is why old lead pipes and lead sheet on roofs visibly sag and thicken at the bottom over decades. Lead creeps at room temperature and steel effectively does not.

So the answer to "is there a threshold" is: at a given temperature there is a stress below which the rate becomes so small that other failure mechanisms will get there first, by an enormous margin. Not a true threshold, but one for practical purposes.

Your intuition is right, and the phenomenon has a name: creep. A material under sustained load deforms slowly over time even well below the stress that would break it immediately, and given long enough that deformation ends in failure.

The mechanism is that atoms in a crystal are not locked in place. They jump between positions occasionally, driven by thermal energy, and under load the jumps that relieve the stress are slightly more likely than the ones that oppose it. Each jump is negligible; over enough time they add up to measurable stretching, then to thinning, then to rupture.

Rates in low-stress conditions are strikingly small — laboratory tests on steel report elongation measured in tens or hundreds of micrometres per year on specimens a few centimetres long. Small, and not zero, and that is the whole point of your question.

One familiar example that makes creep concrete: a bolt that was tightened correctly and is loose years later, with nothing having vibrated.

That is relaxation, which is creep's close relative — the same atomic mechanism, seen as a decaying force at constant stretch rather than increasing stretch at constant force. It is why critical joints get retorqued on a schedule, and why some fasteners are designed to hold their tension over time.

It is happening in ordinary structures around you, slowly, all the time.