We present Chandra and XMM-Newton X-ray data of NGC 5253, a local starbursting dwarf elliptical galaxy, in the early stages of a starburst episode. Contributions to the X-ray emission come from discrete point sources and extended diffuse emission, in the form of what appear to be multiple superbubbles, and smaller bubbles probably associated with individual star clusters. Chandra detects 17 sources within the optical extent of NGC 5253 down to a completeness level corresponding to a luminosity of 1.5 x 10(37) erg s(-1). The slope of the point-source X-ray luminosity function is -0.54 +/- (0.21)(0.16), similar to that of other nearby dwarf starburst galaxies. Several different types of source are detected within the galaxy, including X-ray binaries and the emission associated with star clusters. Comparison of the diffuse X-ray emission with the observed Halpha emission shows similarities in their extent. The best spectral fit to the diffuse emission is obtained with an absorbed, two-temperature model giving temperatures for the two gas components of similar to0.24 and similar to0.75 keV. The derived parameters of the diffuse X-ray emitting gas are as follows: a total mass of similar to1.4 x 10(6) f(1/2) M-circle dot, wheref is the volume-tilling factor of the X-ray emitting gas, and a total thermal energy content for the hot X-ray emitting gas of similar to3.4 x 10(54) f(1/2) erg. The pressure in the diffuse gas is P/k similar to 10(6) f(-1/2) K cm(-3). We find that these values are broadly commensurate with the mass and energy injection from the starburst population. Analysis of the kinematics of the starburst region suggests that the stellar ejecta contained within it can escape the gravitational potential well of the galaxy, and pollute the surrounding intergalactic medium.