Is the following a gerund or participle in this sentence?

I'm not feeling myself today.

This is a "progressive" or "continuous" construction, where a finite inflected form of the auxiliary BE is followed by an -ing form of the verb FEEL.

I think that currently, in approaches to English grammar that treat the gerund as a distinct form from the -ing participle, the -ing form in this construction is generally categorized a participle (the "present", "progressive", or "active" participle).

Its historical origin appears to be more arguable: äüö's answer to the prior question progressive forms: participle or gerund? cites a source that argues that there were originally two distinct constructions, one with BE + a present participle, and one with BE + preposition + gerund, that ended up being conflated in Middle English.

The Cambridge Grammar of the English Language (Huddleston and Pullum, 2002) does not treat the gerund and present participle as distinct inflected forms: it argues for calling the -ing form of verbs the "gerund-participle" rather than using separate terms "gerund" and "present participle" depending on the use.

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BE does not take an object. You might be confusing the concepts of "object" and "complement". A complement is, roughly, some element that "completes" the meaning of a phrase; objects are just one type of complement. When the complement of a verb is a noun phrase, that noun phrase is called the "(direct) object" of the verb; but verbs can take other types of complements, such as preposition phrases ("I am in the garden") or adjective phrases, as in your example ("I'm not happy").