The Large Magellanic Cloud (LMC) offers a unique laboratory to study the diffuse interstellar bands (DIBs) under conditions that are profoundly different from those in the Galaxy. DIB carrier abundances depend on several environmental factors, in particular the local UV radiation field. In this paper we present measurements of twelve DIBs in five lines of sight to early-type stars in the LMC, including the 30 Doradus region. From the high resolution spectra obtained with VLT/UVES we also derive environmental parameters that characterise the local interstellar medium (ISM) in the probed LMC clouds. These include the column density components (including total column density) for the atomic resonance lines of Na I, Ca II, Ti II, K I. In addition, we derive the H I column density from 21 cm line profiles, the total-to-selective visual extinction RV and the gas-to-dust ratio N(H I)/A(V). Furthermore, from atomic line ratios we derive the ionisation balance and relative UV field strength in these environments. We discuss the properties of the LMC ISM in the context of DIB carrier formation. The behaviour of DIBs in the LMC is compared to that of DIBs in different local environmental conditions in the Milky Way. A key result is that in most cases the diffuse band strengths are weak (up to factor 5) with respect to Galactic lines of sight of comparable reddening, EB-V. In the line of sight towards Sk -69 223 the 5780 and 5797 angstrom DIBs are very similar in strength and profile to those observed towards HD 144217, which is typical of an environment exposed to a strong UV field. From the velocity analysis we find that DIB carriers (towards Sk -69 243) are better correlated with the ionised species like Ca II than with neutrals (like Na I and CO). The most significant parameter that governs the behaviour of the DIB carrier is the strength of the UV field.