In this paper, the first of a two-part work, we present the reconstruction and measurement of muon events detected underground by the MACRO experiment at Gran Sasso ${(E}_{\ensuremath{\mu}}>~$ 1.3 TeV in atmosphere). The main aim of this work is to discuss the muon multiplicity distribution as measured in the detector. The data sample analyzed consists of $4.4\ifmmode\times\else\texttimes\fi{}{10}^{6}$ muon events, of which $\ensuremath{\sim}$ 263 000 are multiple muons, corresponding to a total live time of 5850 h. In this sample, the observed multiplicities extend above ${N}_{\ensuremath{\mu}}$=35, with intermuon separations up to 50 m and beyond. Additional complementing measurements, such as the inclusive muon flux, the angular distribution, and the muon separation distribution (decoherence), are also included. The physical interpretation of the results presented here is reported in the following companion paper.