Bose-Einstein condensation (BEC) is a quantum phenomenon, where a macroscopic number of bosons occupy the lowest energy state and acquire coherence at low temperatures. It is realized not only in superfluid 4He and dilute atomic gases, but also in quantum magnets. In three-dimensional (3D) antiferromagnets, an XY-type long-range ordering (LRO) occurs near a magnetic-field-induced transition to a fully polarized state (FP) and has been successfully described as a BEC in the last few decades. An attractive extension of the BEC in 3D magnets is to make their two-dimensional (2D) analogue. For a strictly 2D system, it is known that BEC cannot take place due to the presence of a finite density of states at zero energy, and a Berezinskii-Kosterlitz-Thouless (BKT) transition may instead emerge. In a realistic quasi-2D magnet consisting of stacked 2D magnets, a small but finite interlayer coupling stabilizes marginal LRO and BEC, but such that 2D physics is still expected to dominate. A few systems were reported to show such 2D-limit BEC, but at very high magnetic fields that are difficult to access. The honeycomb S = 1/2 Heisenberg antiferromagnet YbCl3 with an intra-layer coupling J ~ 5 K exhibits a transition to a FP state at a low in-plane magnetic field of Hs = 5.93 T. Here, we demonstrate that the LRO right below Hs is a BEC but close to the 2D-limit, marginally stabilized by an extremely small interlayer coupling J⊥. At the quantum critical point Hs, we clearly capture 2D-limit quantum fluctuations as the formation of a highly mobile, interacting 2D Bose gas in the dilute limit. A much-reduced effective boson-boson interaction Ueff as compared with that of prototypical 3D system clearly indicates the presence of a logarithmic renormalization of interaction unique to 2D. The old candidate for a Kitaev quantum spin liquid, YbCl3, is now established as an ideal arena for a quantum critical BEC in the 2D limit.