Only partly, and not as a website. Squarespace's own export creates an .xml file meant for importing into WordPress (Squarespace Help Center, checked 29 September 2026). It carries layout pages, one blog with its posts, and text and image blocks; album, index, portfolio and store pages, page headers and footers, style settings and custom CSS do not export.
So Squarespace's export is a content transfer, not a copy of your design. To keep the site as it looks, export the published pages instead: this exporter renders each page in a browser and saves its HTML, CSS, JavaScript and assets, ready for any static host. Commerce, Member Areas, Scheduling and form processing run on Squarespace's servers and still need a replacement.