Visiting EU as British citizen long-term

Even if Relaxed is completely right with his answer, I'll try to simplify some issues and add a few other points you should consider:

As a citizen of an EEA country, you have the unlimited right to stay and work in other EEA countries, but you have to fulfil some requirements. Some of the requirements may be covered by national law and differ between the EEA countries. There is also no clear definition of the difference between visitors and residents, you may however end up in situations where the difference do matter.

Since your plan sounds like some kind of vacation, I assume that you have sufficient money to support yourself for the duration of your trip. One of the basic limitations to the right of free movement within the EEA is that you should not end up as a burden for the social welfare system in a visiting country. If you have sufficient savings and are able to live from that, this is not an issue.

Different from country to country is how long you are allowed to stay before you are required to register as a resident. Note here, that the country you are visiting cannot deny the registration if you can support yourself, you may however be required to do some paperwork and visit some kind of public office. For this issue, it may even matter if you live in a hotel (which may do the registration for you) or if you live in some kind of private lodging or in a rented appartment.

The UK health insurance provided by NHS will cover you in other EEA countries as well, but you may have to pay for any required medical services up front and get a refund later from the NHS and the coverage is limited to people 'ordinarily resident' in the UK. There is unfortunately no clear definition of this term in UK law, so in case of a dispute, the NHS may argue that you are not 'ordinarily resident' in the UK after staying abroad for such a long time and deny any compensation for medical expenses. To avoid this problem, you may want to consider getting an additional medical insurance for expats, which covers the countries you are planning to visit.

British citizens are certainly not limited to 90 days in any 180-day period. Before I explain the rules that do apply to you, let's consider where the 90-days-in-any-180-day-period limit comes from, namely article 5 of the Schengen Borders Code:

Entry conditions for third-country nationals

1. For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following:

[…]

“Third-country nationals” means everybody who isn't an EU (technically EEA/Swiss) citizen. This rule explicitly does not apply to you.

Furthermore, the same regulation also provides that:

Scope

This Regulation shall apply to any person crossing the internal or external borders of Member States, without prejudice to:

(a) the rights of persons enjoying the right of free movement under Union law; […]

As a British citizen, you are “a person enjoying the right of free movement under Union law” and those free movement rules trump any Schengen rule.

There is however a kind of 90-day limit that does apply to you, found in particular in directive 2004/38/EC, whose article 6 provide that

1. Union citizens shall have the right of residence on the territory of another Member State for a period of up to three months without any conditions or any formalities other than the requirement to hold a valid identity card or passport.

That's as comprehensive as it gets. It means that if you intend to stay less than three months in any EU country, you can do it without fulfilling any special requirement beside being able to prove you are a British citizen (i.e. holding a passport). At most, you might have to report your presence to the authorities, as explained on europa.eu.

By contrast, article 7 reads:

1. All Union citizens shall have the right of residence on the territory of another Member State for a period of longer than three months if they:

(a) are workers or self-employed persons in the host Member State; or

(b) have sufficient resources for themselves and their family members not to become a burden on the social assistance system of the host Member State during their period of residence and have comprehensive sickness insurance cover in the host Member State; or

(c)

— are enrolled at a private or public establishment, accredited or financed by the host Member State on the basis of its legislation or administrative practice, for the principal purpose of following a course of study, including vocational training; and

— have comprehensive sickness insurance cover in the host Member State and assure the relevant national authority, by means of a declaration or by such equivalent means as they may choose, that they have sufficient resources for themselves and their family members not to become a burden on the social assistance system of the host Member State during their period of residence; or

(d) are family members accompanying or joining a Union citizen who satisfies the conditions referred to in points (a), (b) or (c).

That's one of the places where you can find a reference to being a worker, etc. The way this works is that if you are a worker (or student or have sufficient resources) then you are definitely entitled to stay, possibly after registering yourself in some way.

But even if you don't fulfill any of the conditions, there is a twist. You cannot be detained or forcibly removed and usually you cannot even be fined merely because you are present for more than 90 days on the territory of a member state. What this article means is that the member state can then ask you to leave, which is much more difficult under article 6.

Some countries also offer more generous conditions and most do not actively enforce these requirements so the 90-day limit only becomes relevant when you live somewhere and need to complete some formalities, like getting some personal tax number, opening a bank account, etc. You might then be asked for a proof of address (unlike the UK, many countries have a mandatory registration system) and to show that you have the right to reside in the country.

Those rules are especially relevant when trying to get welfare benefits (cf. the “not [to] become a burden on the social assistance system” stipulation under article 7(1)(b)). So if you don't need to register and don't attract attention to yourself say by applying for some benefits, nobody will ask you anything. You don't have a right to stay in the country but until you hit a point where you have to prove you qualify, it's not illegal to do it either.

Incidentally, the threshold applies to one country at a time, so 90 days in Greece followed by 90 days in Cyprus followed by 90 days in Bulgaria, etc. are OK in any case.