Short story about a Martian invasion . . .
"The Martian Shop", a novelette by Howard Fast, better known for his historical novels such as Spartacus; first published in The Magazine of Fantasy and Science Fiction, November 1959, available at the Internet Archive. Does any of these covers look familiar?
. . . that starts with a mysterious set of stores in major cities
It had been established that three stores had been opened on the same day and the same hour; and more than that, as an indication of a well-organized and orderly mind, the space for each of the stores had been rented on the same day, the leases signed on the same hour. The store in Tokyo was located in the very best part of the Ginza. [. . .] The store in Paris was, of course, on Faubourg St. Honoré. [. . .] The third store was on Fifth Avenue in New York City.
[. . . .]
On Fifth Avenue, as in the other cases, the center focus of the decorating scheme was the crystal replica of the Planet Mars, which was suspended from the ceiling in each shop, and which revolved at the same tempo as Mars itself. It has not yet been determined what type of mechanism activates these globes. The globes, which display a unique and remarkable map of Mars' surface, were installed by the principals, after Trevore had completed the overall alteration and decoration. While the Fifth Avenue storefront is striking, it was done with the type of expensive modesty that would do credit to Tiffany's. The last thing installed was the name of the shop itself, MARS PRODUCTS, in gold letters, each letter a half-inch in relief and five inches high. It has since been determined that these letters are cast out of solid gold.
The story is about these mysterious stores that sell products that do not break and make life easier.
There were four products in the shops: a clock, an adding machine, an outboard motor, and a music box:
The adding machine was a black cube, measuring slightly more than six inches. The covering is of some as yet undetermined synthetic or plastic, inlaid with the curious hieroglyphs that have come to be known as the Martian script, the hieroglyphs in white and gold. This machine is quickly and easily adjusted or sensitized to the sound of an individual voice, and it calculates on the basis of vocal instruction. The results emerge through a thin slit in the top, printed on paper similar to that mentioned before. Theoretically, such a calculator could be built today, but, so far as we know, by only two shops, one in Germany and the other in Japan, and the cost would be staggering; certainly, it would take years of experimental work to develop it to the point where it would deal with thirteen digits, adding, subtracting, multiplying and dividing entirely by vocal command.
The outboard motor was an object about the size of a small electric sewing machine, fabricated of some blue metal and weighing fourteen pounds, six ounces and a fraction. Two simple tension clips attached it to any boat or cart or car. It generated forty horsepower in jet propulsion, and it contained, almost microcosmically, its own atomic generator, guaranteed for one thousand continuous hours of operation. Through a muffling device, which has so far defied even theoretical solution, it produces less sound than an ordinary outboard motor. In each shop, this was explained, not as a muffling procedure, but as a matter of controlled pitch beyond the range of the human ear. Competent engineers felt that this explanation must be rejected.
In spite of the breathtaking implications of this atomic motor, it was the music box that excited the most attention and speculation. Of more or less the same dimensions as the adding machine, it was of pale yellow synthetic, the hieroglyphs pricked out in dark gray. Two slight depressions on the top of this box activated it, a slight touch of one depression to start it, a second touch on the same depression to stop it. The second depression, when touched, changed the category of the music desired. There were twenty-two categories of music available—symphonic music in three chronological sections, chamber music in three sections, piano solo, violin solo with and without accompaniment, folk music for seven cultures, operatic in three sections, orchestra, full cast and orchestra, that is the complete opera, and selected renderings, religious music, divided into five religious categories, popular songs in national sections, instrumental music in terms of eighty-two instruments, jazz in five categories and three categories of children's music.
The salespeople in each of the three shops claimed that the music box had a repertoire of eleven thousand and some odd separate musical selections, but this, of course, could not be put to the test, and varying opinions on this score have been expressed.
The population becomes dependent on these products.
No. The products are displayed and demonstrated in the shops, and orders are taken, but no payments are accepted, no products are delivered, and the shops close after one week.
It was part of a collection, I want to say that the collection was from the late fifties and early sixties.
You may have read it in Howard Fast's 1961 sci-fi collection The Edge of Tomorrow.
I feel like all of the stories in the collection were "Marxist".
Howard Fast was in fact a Communist. From Wikipedia:
In 1943, he joined the Communist Party USA and in 1950, he was called before the House Committee on Un-American Activities; in his testimony, he refused to disclose the names of contributors to a fund for a home for orphans of American veterans of the Spanish Civil War (one of the contributors was Eleanor Roosevelt), and he was given a three-month prison sentence for contempt of Congress.
[. . . .]
In 1952, Fast ran for Congress on the American Labor Party ticket. During the 1950s he also worked for the Communist newspaper, the Daily Worker. In 1953, he was awarded the Stalin Peace Prize. Later that decade, Fast broke with the Party over issues of conditions in the Soviet Union and Eastern Europe.