I wanted to know if there exists a constant with a unit meter-time or say length-time. Dimensionally [LT]. I have searched browsed a lot. Is there any quantity arising with such a unit?
One can probably always derive some contrived example.
If I characterize flow in a pipe by the property of "residence" $R$, for example, where the time required for some slug of liquid to pass through a pipe of length $L$ and cross-sectional area $A$ is $t=RL/A$, then this property would have units of meter-second.
If I wished to characterize the predominance of some linear commodity over time, then I might evaluate the length multiplied by the time it's been used: "No. 12 AWG copper wire has predominated in U.S. house construction since electrification, with a total extent of use of 2.4 trillion meter-years."
Not a fundamental one. See list here.