NGC 4736 is a nearby Sab spiral galaxy, hosting one of the closest examples of a LINER nucleus. We have utilized recent observations by ROSAT and ASCA to characterise the X-ray properties of this galaxy. Twelve discrete X-ray sources are detected within the region subtended by its optical disk, the majority of which are likely to be X-ray binaries associated with the galaxy. By far the brightest source in the X-ray band is positionally coincident with the nucleus of the galaxy and is spatially resolved into a component with a radial extent of ~ 3 kpc plus a point-like core. The broad band (0.1-10 keV) spectrum of this nuclear source is composed of a hard continuum with a spectral slope characteristic of that observed in classical Seyfert nuclei (i.e. power-law photon index, Gamma ~ 1.7), with thermal emission (kT = 0.1-0.6 keV) dominant below 2 keV. An Fe K alpha line may also be present at ~ 6.8 keV. There is no evidence for X-ray temporal variability on timescales of hours to years. A plausible model is that the hard continuum originates in a near-quiescent active galactic nucleus (with L_x ~ 6 x 10^39 erg/s, 0.5-10 keV) embedded in the LINER at the centre of NGC 4736. However, an alternative explanation, namely that the LINER is the site of a dense population of X-ray binary sources, cannot be completely excluded. ; Comment: 11 pages including 4 embedded figures, uses graphicx.sty and mn.sty, accepted for publication in MNRAS