This paper investigates the energy efficiency of massive unsourced random access~(URA) in multiple-input multiple-output quasi-static Rayleigh fading channels. Specifically, we derive achievability and converse bounds on the minimum required energy-per-bit under the per-user probability of error constraint, where the converse bounds contain two parts: one is general and the other is a weaker ensemble bound. Numerical evaluation shows that the gap between our achievability and converse bounds is less than $5$~dB in the considered regime. Some practical schemes are energy-inefficient compared with our bounds especially when there are many users. Moreover, we observe that in contrast to the sourced random access paradigm, the URA paradigm achieves higher spectral efficiency.