Barcelona rains rarely and heavily, usually in short intense bursts in spring and autumn. A rainy day is generally a rainy few hours, and knowing where to go saves it.
Underground
MUHBA at Plaça del Rei — you descend below the square into the excavated Roman city: streets, workshops, a laundry, a wine-making facility, and fish-salting vats, preserved beneath the Gothic Quarter.
One of the largest underground Roman excavations in Europe, and it takes a couple of hours.
Montjuïc's museums
MNAC — the Romanesque frescoes cut from Pyrenean church walls and reinstalled in reconstructed apses. One of the most important collections of medieval wall painting anywhere.
The Gothic and Modernisme collections upstairs, and a building large enough to fill an afternoon.
The Miró Foundation nearby, in a building designed for it.
Both are reached by funicular and cable car, which keeps you out of the rain.
The Maritime Museum
In the Drassanes Reials — medieval royal shipyards, a Gothic hall of enormous stone arches where galleys were built.
The collection includes a full-size replica galley. The space alone justifies the visit.
The other indoor options
The Picasso Museum, in five connected medieval palaces in El Born.
The Palau de la Música Catalana on a guided tour, or better, a concert.
CosmoCaixa, the science museum, with a flooded Amazon section — the best rainy-day option with children.
MACBA and CCCB in El Raval, adjacent to each other.
The Frederic Marès Museum — a collector's accumulation of sculpture and everyday objects, strange and quiet.
The market bars
The genuinely good rainy-day answer.
Santa Caterina and Sant Antoni — covered markets with bars serving excellent food at counter prices, used by residents.
A long lunch at a market counter while it rains outside is a better use of the afternoon than most museums.
La Boqueria is covered too and is priced for the Ramblas.
The arcades and interiors
Santa Maria del Mar, free outside guided hours, and the interior is the point — fourteenth-century Gothic of extraordinary austerity.
The cathedral cloister, with its geese.
The Sagrada Família, which if you have a booked slot is an indoor visit and is arguably better in changeable light.
If it clears
Barcelona drains fast. The streets flood briefly and then run clear, and the light after a storm over the sea is the best the city gets.
Bunkers del Carmel or the Montjuïc terraces immediately afterwards.
Practical
Most museums close Mondays, which is worth checking against a forecast.
Several municipal museums are free on Sunday afternoons and the first Sunday of the month.
The metro reaches all of the above and keeps you dry between them.