Alternatives to NCBI BLAST during US government shutdowns?
b.nota is correct - Just further adding to his answer. The International Nucleotide Sequence Database Collaboration (INSDC) is a consortium between DNA Data-bank of Japan (DDBJ), EMBL-EBI and NCBI. Contributions to each of the 3 databases are shared on daily basis. 

(Thanks to @KonradRudolph) Since EMBL-EBI is not directly accountable to any single government either financially or in terms of research directions so a (temporary) shutdown equivalent to that of the NCBI is vastly less likely. Rather, like CERN, it’s an international treaty organization and thus probably a lot more robust.  

BLAST

Answering part 2 of your question: YES! BLAST is available in both DDBJ and EMBL-EBI on following links:


BLAST - DDBJ
NCBI BLAST+


Scope

Here is what INSDC's website says about its scope:


  INSDC covers the spectrum of data raw reads, though alignments and
  assemblies to functional annotation, enriched with contextual
  information relating to samples and experimental configurations.


Following table (taken from INSDC website) may help you have better understanding about respective data types comparison for 3 contributors: