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I work remotely and thus I can live anywhere while doing my job. As a French citizen, what are countries where I can easily reside for 6 months or more, without needing a visa or where one can easily be obtained (without meeting specific conditions).
(I realize that this might be a difficult question to fully answer. Feel free to close if it is too broad, although it is a question addressing a genuine problem.)
As an EU citizen you can live long term in any other EU country. For less than three months you don't need to do anything (apart form having your passport or National ID with you). For longer terms however you have to either be engaged in economic activity (work there), or have enough resources to not become a burden on the social services of the host Member State. As you said you have a steady income, this point should apply to you. More information
While Switzerland, Iceland, Lichtenstein and Norway are not part of the EU, the above also applies for them. You might to register yourself with local authorities (like in Norway or Iceland), and show them you are self supporting.
As countries outside of the EU don't recognize the EU as one, but as separate countries, it depends on your home country where you might live struggle free for a longer term. While this example is for short stay it highlights the issue clearly: most EU countries are part of the Visa Waiver Program, meaning they can enter the USA (for short term stay) easier, but some are not. For example as a French citizen you can stay visa free in Canada, Peru, Mexico and Panama for 180 days (but require visa for longer stay).
(Note: the question was asked generally about EU citizens in the first place)
Outside of EU/EEA/Switzerland, mentioned by SztupY above, one of the few countries with a truly liberal visa policy is Georgia:
You can stay in the country for up to 365 days and by traveling elsewhere once a year you can essentially stay there forever.
Purchasing a piece of real estate (which can be as low as 10000 euros for a small apartment), allows you to apply for permanent residency.
It's even possible to work on a tourist visa (according to information from 2014), although the salaries are most likely mediocre for the average Frenchman.
