The Fourdrinier Machine was invented in 1798 and by the 1830s would have been common in the paper making industry (were large spools of paper are used for commercial printing e.g. newspaper presses).

Paper was originally made form cotton or linen rags, and wasn’t until 1827 where an accidental discovery led to cheaper options for manufacturing paper.

With the paper machine it seems anything was possible. In 1827 a farmer in Meadville, PA was making potash and taken to lining the hopper with long straw that became crushed in the process. He noticed the macerated pulp may have applications in paper-making, and so carted a small load over to the nearby paper mill owned by John Shyrock where it showed great promise. Straw has only 35 percent cellulose content, but when mixed with rags the paper gained a durability enough for use as wrapping or sack paper.

The 1830s would have been the first time that paper in this form was more readily available. This is just speculation, but I believe that you would order directly from the paper mill.

During the 1830s the paper industry worked in perfect harmony as hand mills turned out superior articles of laid writing papers, while the new machine mills made all the inexpensive stock as book, news, wrapping, and wall paper.

Quotes from “Changes in Print Paper During the 19th Century” by AJ Valente: https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1124&context=charleston

A spool of paper that’s 1,000 feet long is also probably a lot smaller than you’re imagining- more like an extra large roll of wallpaper.

There’s a paper spool calculator here https://www.handymath.com/cgi-bin/rollen.cgi, and a 6 inch diameter roll of paper with a 1 inch hole in the middle has over 1,000 feet of paper if it’s 0.05mm thick.

Luckily for Babbage, the paper he needed had become widely available a few years before. In the first two or three decades of the 19th century, the process of papermaking went from being the same process used to make the paper Gutenberg used to being more-or-less the same process used today.

In 1800, paper was almost entirely made by hand, in sheets of various sizes, from linen rags (and cotton, though it's lower quality). Paper is made by soaking, pounding, and fermenting rags in water until they became a slurry of the constituent fibers, and then scooping that slurry into mesh frames the size of the desired sheet of paper. The paper was then pressed between sheets of felt to dry.

A small "industrialized" mill (using water power) could produce perhaps 2000 sheets of paper a day (4 reams) while a hand mill could do perhaps 1000. Each sheet had to be relatively small due to the challenges of physically lifting and scooping the paper pulp from the vat. (See this video to see why bigger sheets would be so difficult). The largest legal size of paper in British tax records in 1781 is 38.5 inches by 26. In this era, half the cost of a book could be the paper for printing.

By the 1830s, however, the hand method had been largely to entirely supplanted by an entirely mechanical, continuous process. Using a continuous roll of mesh, the pulp is drained onto a mesh conveyor belt, pressed between rolls of felt and fed out as a continuous roll. This could then be cut to size as desired. There were two designs developed around this time; the eventual "winner" was the so-called Fourdrinier machine (named after its financial backers, not its inventor). By 1811, there were 18 Fourdrinier machines in the UK. As for how long these rolls could be, an American papermaker, operating from a knockoff of the runner-up design (there were export controls on papermaking machines placed by the British government), exhibited a thousand-foot roll of paper at a meeting of the American Philosophical Society in Philadelphia in 1817.

That addresses at least the availability of a thousand-foot continuous roll of paper. But what about cost?

The material cost of rags was already high and had been climbing for a long time--more literacy over the course of the 18th century means more demand for reading material and more demand for paper. So, by the 1830s we find ourselves at the tail end of the age of rag, but we have not yet entered the age of wood (wood pulp began to be used widely in the 1860s). Instead, it's basically the age of throwing spaghetti at the wall and seeing what sticks.

Various enterprising paper mill owners used a wide variety of other types of fibers which met with a wide varieties of levels of success. Straw was widely used. Manila, made with hemp rags and rope scraps (rope was made with abaca, a fibrous banana plant from the Philippines, hence the name), was invented in 1843. Old sails and jute were also used, as well as raw cotton and flax scraps from mills, and recycled paper. Almost all of these were only used as wrapping paper or paper sacks. By 1870, the list of materials tested for use as a rag substitute ran for two full pages, and included moss, algae, leather, ivory, reeds, and the wrappings from Egyptian mummies. Of these, straw was probably the most common.

None of this is likely to have been actually used by Babbage, but the creation of various types of low-quality wrapping papers allowed prices to come down on writing papers, as the rag they were made of was no longer also being used for these other types of paper. Unfortunately, I can't give you an exact number and I can't give you anything specifically British which may be different due to different economies and tax structures, but I can give you at least a sense of the changes in price over the course of this era with some data from Massachusetts. In 1804 one full sheet (so perhaps 2x3 feet at most) of low-quality writing paper cost 25 cents. In 1839, it cost 10 cents. Of course, by 1839 that writing paper was being cut off of a roll made by a Fourdrinier machine. So, a roll 1000 feet long and 6 feet wide would make 1000 sheets of 2x3 foot paper, making the retail value of that roll $100.00.

We are getting very r/theydidthemath here, and 19th-century conversion rates are sketchy ($100 in 1839 is perhaps around 20 pounds sterling) so I'll leave it at that, but of course Babbage wouldn't have put a 6-foot wide roll of paper on the train, and it's unlikely he would have paid full retail price for what he did buy--since he purchased what amounts to a half-manufactured product.

In short, he would have been able to buy it, and for the wealthy son of a banker it wouldn't have been too expensive (Babbage inherited 100,000 pounds in 1827).

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Sources:

"Changes in Print Paper During the 19th Century," by AJ Valente, 2010

"Notes on Eighteenth-Century British Paper," by Philip Gaskell, in The Library: Transactions of the Bibliographical Society. Fifth Series, v. 12, no. 1, March 1957

Paper: Paging Through History, by Mark Kurlansky, 2016

Paper Conservation Catalog, by the American Institute for Conservation Book and Paper Group. Chapter 4: "Support Problems".

A History of Paper Manufacturing in the United States, 1690-1916, by Lyman Horace Weeks, 1916

A Chronology of Paper and Paper-Making, by Joel Munsell, 1870

Comparative wages, prices, and cost of living (from the Sixteenth annual report of the Massachusetts Bureau of Statistics of Labor for 1885), Carroll D. Wright.

An Encyclopedia of the Book by Geoffrey Ashall Glaister, 1960