use MagentoFrameworkApiFilterBuilder;
use MagentoFrameworkApiSearchFilterGroupBuilder;
use MagentoFrameworkApiSearchCriteriaBuilder;
use MagentoCmsModelBlockRepository;

/**
 * @var FilterBuilder
 */
private $filterBuilder;

/**
 * @var FilterGroupBuilder
 */
private $filterGroupBuilder;

/**
 * @var SearchCriteriaBuilder
 */
private $searchCriteriaBuilder;

/**
 * @var BlockRepository
 */
private $blockRepository;

public function __construct(
    FilterBuilder $filterBuilder,
    FilterGroupBuilder $filterGroupBuilder,
    SearchCriteriaBuilder $searchCriteriaBuilder,
    BlockRepository $blockRepository
) {
    $this->filterBuilder = $filterBuilder;
    $this->filterGroupBuilder = $filterGroupBuilder;
    $this->searchCriteriaBuilder = $searchCriteriaBuilder;
    $this->blockRepository = $blockRepository;

}

$filter = $this->filterBuilder->setField('identifier')
    ->setConditionType('eq')
    ->setValue('men-block')
    ->create();

$filter2 = $this->filterBuilder->setField('identifier')
    ->setConditionType('eq')
    ->setValue('gear-block')
    ->create();

$filterGroup = $this->filterGroupBuilder->setFilters([$filter, $filter2])
    ->create();

$filter3 = $this->filterBuilder->setField('is_active')
    ->setConditionType('eq')
    ->setValue(1)
    ->create();

$filterGroup2 = $this->filterGroupBuilder->setFilters([$filter3])
    ->create();

$searchCriteria = $this->searchCriteriaBuilder->setFilterGroups([$filterGroup, $filterGroup2])
    ->create();
$blockItems = $this->blockRepository->getList($searchCriteria)->getItems();

SELECT `main_table`.* FROM `cms_block` AS `main_table` WHERE ((`identifier` = 'men-block') OR (`identifier` = 'gear-block')) AND ((`is_active` = 1))