For an interview this is a good question. If this was asked as an actual task in your job, then there would be a simple answer of use a third party component, there are a few good commercial ones.
While we can't say for sure what your interviewer wanted, for me this is a good question precisely because it is so open ended and has so many correct possible answers.
You can talk about the UI and how to implement the kind of dynamic grid you need for a spreadsheet and all the functionality of the cells and rows and columns and selection of cells and ranges and editing of values and formulas. You probably could talk for a while on the UI implications alone.
Alternatively you can go the data route, talk about data structures to hold a spreadsheet, talk exactly about links between cells for formulas, talk about how to detect and deal with circular references, talk about how in a browser you have less control over memory and for very large spreadsheets you could run into problems earlier. You can talk about what is available in JavaScript vs a native language and how this impacts the data structures and calculations. Also along with data, a big important issue with spreadsheets is numerical accuracy and floating point number calculations. Floating point numbers are made to be fast but are not necessarily accurate in extreme levels of precision and this leads to a lot of confusing questions. I believe very recently Excel switched to their own representation of a fixed decimal number as it's now viable to due spreadsheet level calculations without using the built-in floating point calculations. You can also talk about data structures and calculation and how they affect performance. In a browser you don't have threads (yet) so you can't run all the calculations in the background. If you have 100,000 rows with complex calculations and change one value that cascades across everything, you can get a warning about a slow script. You need to break up the calculation.
Finally you can run form the user experience angle. How is the experience in a browser different from a native application? What are the advantages and what cool things can you do in a browser that may be difficult in a desktop application? What things are far more complicated or even totally impossible (example, associate your spreadsheet app with a file type so a user can double-click a file and open it in your online spreadsheet app, although I may be wrong about that still being unsupported).
Good question, lots of right answers, very open ended.
On the other hand, you could also have had a bad interviewer that is specifically looking for the answer they want and in that case you're pretty much out of luck unless you're telepathic.