Can close binaries have a very eccentric orbit? For example, are there binaries with an orbital period &lt; 5 day  and that are very eccentric?

What kind of factors could slow down orbital circularisation?

Are there examples of binaries with p&lt;5day and e&gt;0.1?
There are surprisingly many examples of short period, eccentric binry systems.

A very incomplete list would include

KIC4544587 P=2.19 d  e=0.288 +/- 0.026 Hambleton et al. (2013)

HD174884 P=3.66 d  e=0.2939 +/- 0.0005 Maceroni et al. (2010)

CoRoT 102918586 P=4.39 d  e=0.249 +/- 0.005 Maceroni et al. (2013)

HD313926 P=2.27d  e=0.209 +/-0.001 Rucinski et al. (2007)

In each of these cases, the stars are relatively warm, with thin (if any) subphotospheric convection zones. Tidal circularisation is much more effective in stars with thick convection zones (giants, or cooler main sequence stars). Therefore it is possible that all of these objects (and, as I say, there are many other examples) are too young to have circularised.

Some theory is presented by Zahn et al. (2005). Indeed, circularisation times are very short - basically you expect everything with a solar mass or below to be circularised if its period is less than 6 days, but the process is less efficient for higher mass stars. Circularisation will essentilly take place during pre-main-sequence evolution. Since a circular orbit represents the minimum energy configuration, the only way you are going to get non-circular orbits in solar-type and lower mass stars is if they are very young (less than some tens of millions of years), the binary system formed after their pre-main-sequence evolution or if there is an injection of energy via a dynamical interaction or an eccentric third body in a much wider orbit.