You are an:
An exclusive group of people
The ELECT are a small group of people chosen by a much larger group as their representatives.
A city that’s eternal
The 'Eternal City' is a nickname traditionally given to Rome in Italy. However, note that here we need its local name in Italian: ROMA.
A book that’s periodical
A magazine is a periodical, and can be shortened to MAG.
A trap to capture what’s feeble
A NET is a trap.
Put the letters together and you can make:
ELECTROMAGNET - a peculiar attraction indeed (re the title)!