I would definitely keep the store open - yes making all your pages uncrawlable by search engines will negatively impact your SEO.
If you don't want to take sales, you could disable your add to cart button like this:
1) Find the button in your code (usually it's in something like product-form.liquid)
2) Comment out the button:
{% comment %}
Everything in here will not be loaded on the webpage. Wrap your add to cart button in these comment tags.
{% endcomment %}
3) You can replace with a message saying your store is pausing orders until X date.
Oooooh wait I think I have a better idea:
1a) Set all your inventory to 0
2a) Install the Now Back in Stock app, which will show a form instead of the add to cart button (for products with 0 inventory)
3a) Include a message there that orders are paused, but customer can enter their email address to get notified when "back in stock" / "fulfillment service is back in action".
Be well and in good health
Founder of
Speed Boostr (Shopify optimization experts, theme customization, custom app development).
Creator of
Shopify Analyzer (first performance analysis tool for Shopify sites, free for the Shopify community).
Creator of
Order Automator (auto tag orders and customers + auto fulfill orders + email notifications).
Creator of
Theme Scientist (A/B testing app) and
Tip Jar (add a tip button to your store).
