まあその、自慢できる事でもないんだけど。というか、臆面もなく吹聴できる話でもない。

、。、。

Well, it's not exactly something to brag about. Or rather, it's not the kind of story you'd go around loudly broadcasting without any shame.

Literal

Well that, pride-can-do thing [also-でも] not-is-[but-けど]. Rather [than-というか], shame-less-even [object-を] brag-about-can story [also-でも] not-is.

A self-deprecating qualifier opening. 臆面もなく ('without any shame/reserve') is a literary expression — 臆面 literally 'timid face' means 'shame, hesitation'. 吹聴する (ふいちょうする) is a formal verb meaning 'to broadcast, trumpet, spread word loudly'. というか is a colloquial 'or rather' used to walk back what was just said. まあその is a multi-filler hedge opener.