Zhen Lin points out below that I've been way too cavalier with transferring model structures along a reflection. So the following answer is not clearly correct. I will leave this up as community wiki because I still think it addresses the spirit of the question, showing that spaces can be "modeled" in some sense by globular sets (or even just by graphs).

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Contrary to my guess in the comments, the answer is _no_: there does exist a model structure on globular sets (reflexive or otherwise) which is Quillen equivalent to the Kan-Quillen model structure on spaces.

To see this, note that if $\mathcal A$ is a reflective subcategory of $\mathcal B$, and if $\mathcal A$ has a model structure, then the model structure transfers to $\mathcal B$, and the resulting adjunction is a Quillen equivalence.

Now, as mentioned in the comments, the category $Gph$ of graphs (reflexive or otherwise) is a reflective subcategory of $Glob$. Moreover, the category $Pos$ of posets is a reflective subcategory of $Gph$. Thus $Pos$ is reflective subcategory of $Glob$. So it will suffice to find a model structure on $Pos$ which is Quillen equivalent to topological spaces. This is proven by [Raptis](https://dx.doi.org/10.4310/HHA.2010.v12.n2.a7), by transferring the [Thomason model structure](https://ncatlab.org/nlab/show/Thomason+model+structure) on $Cat$.