Can I apply for an eTA with a Refugee Travel Document?

No. Most people using a Refugee Travel Document to travel to Canada will need a visitor visa.

Exceptions:

If you have a valid Canadian Refugee Travel document, you are exempt from both the visa and eTA requirements.
If you’re a lawful permanent resident of the United States and don’t have a passport, you can travel to Canada with your valid U.S. Refugee Travel Document (I-571) and official proof of your lawful permanent resident status.

Complete list of acceptable status documents for lawful permanent residents of the U.S.

You need an official proof of status as a lawful permanent resident of the U.S., such as one of the following:

valid permanent resident card (Form I-551)
foreign passport with an unexpired temporary I-551 stamp (also known as an Alien Documentation, Identification and Telecommunication [ADIT] stamp)
foreign passport with a temporary I-551 printed notation (“Upon endorsement serves as temporary I-551 evidencing permanent residence for 1 year”) on a machine-readable immigrant visa upon endorsement with a U.S. Customs and Border Protection admission stamp
expired permanent resident card (Form I-551) with Form I-797 (Notice of Action) for pending Form I-751 (Petition to Remove Conditions on Residence) or Form I-829 (Petition by Investor to Remove Conditions on Permanent Resident Status)
expired permanent resident card (Form I-551) with Form I-797 (Notice of Action) for pending Form I-90 (Application to Replace Permanent Resident Card [green card])
valid re-entry permit (Form I-327)
Form I-94 with an unexpired temporary I-551 stamp (ADIT stamp) and a passport-style photo