Why does the beryllium 2+ ion have a larger radius than a helium atom?
You are mixing apples and oranges.
Or, to be more precise, an ionic radius for $\ce{Be^2+}$ with coordination number (C.N.) 6 and van der Waals radius of $\ce{He}$.
To make it clear I compiled data for van der Waals and covalent radii [1, p. 9-58] as well as ionic radii [1, p. 12-13]:

$$
\begin{array}{lccccc}
\hline
\text{Element} &amp; R_\mathrm{vdW}/\pu{Å} &amp; R_\mathrm{cov}/\pu{Å} &amp; \text{Ion} &amp; \text{C.N.} &amp; R_\mathrm{i}/\pu{Å} \\
\hline
\ce{Be} &amp; 1.52 &amp; 0.99 &amp; \ce{Be^2+} &amp; 4 &amp; 0.27 \\
  &amp;  &amp;  &amp;  &amp; 6 &amp; 0.45 \\
\ce{He} &amp; 1.40 &amp; 0.37 &amp;  &amp;  &amp; \\
\hline
\end{array}
$$

Apparently, there is no experimental ionic radii for helium to compare with as helium, well, represents one of the most non-reactive elements.
Also, all radii set are determined using various methods and cannot be directly compared.
From the foreword [1. p. 9-58]:


  There are many scales of van der Waals radii, but they are not fully consistent with one another. The van der Waals radii determined by Bondi [...] from x-ray diffraction data, crystal densities, gas kinetic collision cross sections, critical densities, and liquid-
  state properties are the most widely used values. [...]
  
  The covalent radii tabulated here are recommendations for single covalent bonds, 
  and they are based on a comprehensive evaluation of experimental data [...].


Ionic radii are estimated first and foremost from the experimental crystal structure data, averaged and supplemented by empirical and theoretical calculations (extrapolations, e.g. using Zachariasen method).

References


Haynes, W. M.; Lide, D. R.; Bruno, T. J. CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and Physical Data.; 2017; Vol. 97. ISBN 978-1-4987-5429-3.