These polyspiro compounds such as the one you gave are known as [m,n]rotanes, where m is the size of the central ring and n is the size of the decorating rings. So, your compound would be called [5,5]-rotane.
A Reaxys search unfortunately doesn't show anything for [5,5]-rotane, but in general, many such compounds have been made. A lot of the articles are in German or French (there are quite a few such articles). However, I did find one in English: J. Am. Chem. Soc. 1998, 120 (2), 317–328.
The authors synthesised and obtained X-ray structures of a number of rotanes, among them [5,4]rotane, which is the closest to the [5,5] that you are looking for. I quote:
In [5.4]rotane (7), the cyclopentane ring adopts a slightly distorted envelope conformation: C(1), C(5), C(9), and C(17) form an almost perfect plane (mean deviation 0.021 Å), while the flap atom C(13) is located 0.684 Å above this plane. The cyclobutane rings at C(13) and C(9) are nearly planar (φ = 9.7°) and planar (φ = 0.2°), respectively. The remaining three cyclobutane rings are moderately puckered (φ = 23.1, 26.5, and 28.2°).
The torsion angle φ of cyclobutane refers to the angle between the two planes that define the molecule. A completely planar four-membered ring would have φ = 0°, and an increasing value of φ indicates greater deviation from planarity:
As you can see in the X-ray structure, the five attached cyclobutane rings do eclipse each other to a certain extent: the obvious exception is the ring fused to C(13). This is not a very surprising conclusion at all. The envelope conformation of unsubstituted cyclopentane is known to suffer from torsional strain (i.e. unfavourable eclipsing interactions between substituents). Carey and Sundberg's Advanced Organic Chemistry, Part A writes (pp 162–3):
There is minimal angle strain in cyclopentane, but considerable torsional strain is present. Cyclopentane is nonplanar and the two minimum energy geometries are the envelope and the half-chair. [...] The planar portions of both conformations have torsional strain owing to C−H and C−C bond eclipsing.
There's a ton more information in the paper, and I'd recommend giving it a read if you are interested - I just tried to pick out the parts directly relevant to your question here, since you indicated that you were interested in the conformation of such a compound.